206      IX   ANALYTIC FUNCTIONS
PROBLEMS
1.   Let fez"), (bnz") be two power series in one variable, the bn being real and >0;
suppose \imajbn — s.
h-»oo
(a)   Suppose the series (bnzn) is convergent for \z\ < 1, but not for z = 1 (which means

that if ck = 2 ^»» ^m c* = + 00)* Show that the series (anzn) is absolutely convergent
for \z\ < I , and that, if I = [0, 1 [,
lim
z-»l, zel
(Observe that, for any given k,    lim   ( £ £„ zn>) = + oo).
Z-»l,  ZelVn^Jt	/
(b)   Suppose the series (bnzn) is convergent for every z. Show that the series (anzn) is
absolutely convergent for every z, and that if J is the interval [0, -foo[ in R, then
lim
Z-* +QO.Z6J
(Same method.)
oo
(c)	Show that if the series (an) is convergent and £ an — s, then the series (anzn) is
fl = 0
00
absolutely convergent for |z| < 1, and that     lim    J] anzn — s. (Apply (a) with bn ~ I
z-»l, zel n = 0
for every «; this is "Abel's theorem".)
(d)	The power series ((— 1)V) has a radius of convergence 1, and its sum 1/(1 + z)
tends to a limit when z tends to 1 in I, but the series ((—!)") is not convergent (see
Problem 2).
2.   Let (anzn) be a power series in one variable having a radius of convergence equal to 1;
let/(z) be its sum, and suppose that/(I—) exists. If in addition lim nan = 0, show that
h-+oq
the series (an) is convergent and has a sum equal to /(I—). ("Tauber's theorem":
observe that if \nan\ ^eforn^k, then, for any N > k, and 0 ^ x < 1
and
3.   Let (anzn) be a power series in one variable having a radius of convergence r > 0, and
let (bn) be a sequence of scalars ^0 such that <?= lim (bn/bn+1) exists and \q\<r.
Show that, if
cn = aobn + 0i 6n-i H ----- h anbQ
lim (cn/bn) exists and is equal